Estimates for minimal number of periodic points for smooth self-maps of simply-connected manifolds

Description

We consider a closed smooth connected and simply-connected manifold of dimension at least 4 and its self-map f. The topological invariant D_r[f] is equal to the minimal number of r-periodic points in the smooth homotopy class of f. We assume that r is odd and all coefficients b(k) of so-called periodic expansion of Lefschetz numbers of iterations are non-zero for all k dividing r and different from 1. We consider the simplified version of the invariant: D_r[f](mod 1) (which is equal either D_r[f] or D_r[f]+1).

The determination of the exact value of the invariant is a challenging task even for relatively small values of r and low-dimensional manifolds. Nevertheless, in this dataset we are able to provide the estimates for D_r[f](mod 1) for manifolds of dimension between 4 and 21.

We consider an odd r that is a product of 10 prime numbers, each to the power a_i, where 1 ≤ i ≤ 10 and 0 ≤ a_i ≤ 10. In addition r satisfies the following condition: the number of a_i equal to 1 is greater or equal than l (where the dimension of the manifold M is either 2l or 2l+1). The definition of D_r[f](mod 1) and the combinatorial scheme which enables its computation is given in G. Graff and J. Jezierski [J. Fixed Point Theory Appl. 13 (2013), 63-84, https://doi.org/10.1007/s11784-012-0076-1].

The data consists of 9 files: dim4_5.txt, dim6_7.txt, dim8_9.txt, dim10_11.txt, dim12_13.txt, dim14_15.txt , dim16_17.txt, dim18_19.txt, dim20_21.txt each of which contains pairs of lists in the form [a_i: a_i ≠ 0 and 1 ≤ i ≤ 10], [lower_bound, upper_bound], where the first place defines the powers of the prime numbers in the decomposition of r and the second gives the lower and upper bound for the invariant D_r[f](mod 1).

Dataset file

D_r(estimates)-Most-Danych.zip

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Year of publication:

2020

Verification date:

2020-12-17

Dataset language:

English

Fields of science:

mathematics (Natural sciences)

DOI:

10.34808/081x-6069

Funding:

Topological invariants: the structure of a space and its self-maps

Verified by:

Gdańsk University of Technology

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